Lecture 11 : Fixed Income
Zero Coupon Bonds
Consider a risk-free zero-coupon bond
Face value (final payoff) of \(F\) with a maturity of \(T\) years
If the interest rate for maturity \(T\) is \(r_{T}\), then the current price of this bond is:
\[ P_{0} = \frac{F}{(1+r_{T})^{T}} \]
Yield Curve
Notation \(r_{T}\) implies variation in \(r\) over time
Yield curve plots the term structure:
- In theory, produced from zero-coupon bonds or stripped Treasuries
Yield Curve
Notation \(r_{T}\) implies variation in \(r\) over time
Yield curve plots the term structure:
- In theory, produced from zero-coupon bonds or stripped Treasuries
Coupon Bonds
- Coupon bonds pay coupon C in intervening periods (assume annually)
\[ P_{0} = \sum_{t=1}^{T} \frac{C}{(1+r_{t})^{t}} + \frac{F}{(1+r_{T})^{T}}, \]
- \(C\) : interest or coupon payments
- \(T\) : number of periods (years, here) to maturity
- \(r_{t}\) : prevailing discount rate for horizon \(t\)
Yield to Maturity
- Suppose we observe coupon \(C\), face value \(F\) and and price \(P\) of a single bond.
- Yield-to-maturity (YTM) solves for the discount rate that makes this price “right”:
\[ P = \sum_{t=1}^{T} \frac{C}{(1+YTM)^{t}} + \frac{F}{(1+YTM)^{T}}, \]
- YTM is a weighted average of \(r_{1}\) through \(r_{T}\)
- Plots of YTM by horizon approximate the true zero coupon yield curve
Yield to Maturity – Example
- Consider 10 year bond with face value of 100 and coupon payments of 5 dollars per year
- If these bonds trade below their par value, this implies YTM exceeds coupon rate
\[ 92.64 = \sum_{t=1}^{10} \frac{5}{(1+0.06)^{t}} + \frac{100}{(1+0.06)^{10}}, \]
- If bond trades at a premium, then coupon rate is higher than the discount rate:
\[ 108.11 = \sum_{t=1}^{10} \frac{5}{(1+0.04)^{t}} + \frac{100}{(1+0.04)^{10}}, \]
Coupon Bonds and Yield Curves
Note, YTM collapses \(r_{1}\) through \(r_{T}\) to a singular discount rate
Suppose we want to solve for entire yield curve
If we want to know \(T\) interest rates over the yield curve, we need \(T\) bonds (roughly)
- T equations and T unknowns
Once we have interest rates for entire yield curve, can price any risk free bond
Coupon Bonds and Yield Curves
Example:
- Use three bonds to solve for \(r_{1}\), \(r_{2}\), and \(r_{3}\)
\[ P_{1} = \frac{C_{1} + 100}{(1+r_{1})}\\ P_{2} = \frac{C_{2}}{(1+r_{1})} + \frac{C_{2} + 100}{(1+r_{2})^2}\\ P_{3} = \frac{C_{3} }{(1+r_{1})} +\frac{C_{3} }{(1+r_{1})^2} + \frac{C_{3} + 100}{(1+r_{3})^3} \]
Forward Rates
Consider a contract in which we agree to lend money for 1 year, beginning 1 year in the future (repayment 2 years in the future)
- \(f_{1,2}\) is the forward rate for a 1 year loan issued 1 year from today
Generally, we will call \(f_{j,k}\) the forward rate which applies from period \(j\) to \(k\)
- \(f_{0,1}\) is the normal 1 year spot rate, \(r_{1}\)
A comment on notation: we use \(r_{3}\) to mean today’s prevailing interest rate for a 3-year horizon. This is different from BKM (in which \(r_{3}\) refers to the 1-year rate which will prevail three years from now)
Forward Rates
- Now consider two equivalent strategies:
- Buy $1 worth of a two-year zero coupon bond
- Receive \((1+r_{2})^{2}\)
- Buy $1 worth of a one-year zero coupon bond and a forward contract for 1 year beginning 1 year in the future
- Receive \((1+r_{1})(1+f_{1,2})\)
- By no arbitrage, since the bonds and contracts are riskless,
\[ (1+r_{1})(1+f_{1,2}) = (1+r_{2})^{2} \]
- This implies that the 1- and 2-year bond rates set the forward rate:
\[ f_{1,2} = \frac{(1+r_{2})^{2}}{(1+r_{1})} - 1 \]
Forward Rates
- We can extend this to the general case, solving for the forward rate from \(t\) to \(T\):
\[ (1+r_{T})^{T} = (1+r_{t})^{t}(1+f_{t,T})^{T-t}\\ f_{t,T} = \left[\frac{(1+r_{T})^{T}}{(1+r_{t})^{t}}\right]^{\frac{1}{T-t}} - 1 \]
For example, \(r_{1} = 5\%\), \(r_{2} = 6\%\):
\[ f_{1,2} = \left[\frac{(1.06)^{2}}{(1.05}\right] - 1 = 7.01\% \]
Forward Rates and Term Structure
Suppose investors are risk-neutral (or the future is certain)
Then, forward rates should be equal to the expectations of interest rates (why? - what trade could you do if not?)
- \(f_{1,2} = E(r_{1,2})\)
- \(f_{2,3} = E(r_{2,3})\)
Hence, long-term rates will represent a geometric average of current and future expected short rates
\[ (1+r_{2})^{2} = (1+r_{0,1})(1+f_{1,2}) = (1+r_{0,1})E(1+r_{1,2}) \]
Forward Rates and Term Structure
Now what if the investor is risk-averse and investing over two periods
Faces a choice between a sure two period rate, or a roll-over of a short-term bond
Recall that a risk-averse investor prefers a sure thing if the expected returns are equal
In equilibrium, two period investors bid down the long-term rate:
\[ (1+r_{2})^{2} = (1+r_{0,1})(1+f_{1,2}) < (1+r_{0,1})E(1+r_{1,2}) \]
Forward Rates and Term Structure
More commonly, we consider risk-averse one-period investor (needs liquidity at t=1)
Face a similar choice: a sure one-period rate, or a two-period bond that they sell in period 1
- Buy 1-year bond: \((1+r_{1})\)
- Buy 2-year bond and sell at period 1 at price of \((1+r_{2})^{2}/(1+E(r_{1,2}))\)
- Risky position is now flipped
Recall that a risk-averse investor prefers a sure thing if the expected returns are equal
In equilibrium, one-period investors bid down the short-term rate:
\[ (1+r_{2})^{2} = (1+r_{0,1})(1+f_{1,2}) > (1+r_{0,1})E(1+r_{1,2}) \]
Interpreting yield curves
Commonly,upward sloping yield curves are interpreted as evidence of:
- Higher future short rates
- Liquidity preference for short-term bonds
Define the liquidity premium as the difference between the forward rate and the expected future short rate
\[ K = (1+ f_{t,t+1}) - E(1+r_{t,t+1}) \]
The larger this is, the more that you pay for “certainty” of next period’s short-rate
Can generate common yield curve patterns under different expectations and values of K
\(\Delta\) Bond Price and \(\Delta\) Yield to Maturity
Bond Pricing Relationships
- Bond prices and yields are inversely related.
- Long‐term bonds tend to be more price sensitive than short‐term bonds.
- Price sensitivity is inversely related to the bond’s coupon rate.
- Price sensitivity is inversely related to the yield to maturity at which the bond is selling.
- An increase in a bond’s yield to maturity results in a smaller price change than a decrease of equal magnitude.
Duration
Need a single measure that tells us the interest rate sensitivity of a bond/portfolio
Duration is a measure of the effective maturity
The weighted average of the amount of time until each payment is received, with the weights proportional to the present value of the payment
Duration is shorter than maturity for all bonds except zero coupon bonds
Duration is equal to maturity for zero coupon bonds
Duration: Calculation
- Define duration as follows:
\[ D = \sum_{t=1}^{T} t \times w_{t} \\ w_{t} = \left[\frac{CF_{t}}{(1+YTM)^{t}}\right] / P, \]
where YTM is the yield to maturity, \(P\) is the price, and \(CF_{t}\) is the cashflows at time \(t\)
- Can show that
\[ \frac{\Delta P}{P} = -D\frac{\Delta (1+YTM)}{(1+YTM)}. \]
Modified Duration
- Alternatively, modified duration (\(D^{*}\)) divides duration by \((1+YTM)\) such that
\[ \frac{\Delta P}{P} = -D^{*} \Delta YTM \]
Duration Matching
Often, duration is important for investors who want to “immunize”" themselves from interest rate exposure
Consider our Florida Pension Fund case
- Modified duration of liabilities to firefighter is \(\sim\) 9.9 years (taking YTM of 7.75% as given)
- Reflects the weighted average maturity of their obligations
- Note, a 1% decrease in interest rates will increase the value of liabilities by 9.9%
- Why?
Duration Matching
Suppose FL would like to ensure it is hedged against interest rate changes
Invest in a bond portfolio with a duration equal to that of its liabilities
A couple of possibilities:
- A zero coupon bond expiring in 9.9 years
- Buy 25 zero coupon bonds paying 100K, 103K, 106K, in years 1, 2, 3 etc.
What are some issues with those strategies?
Caveats
Duration is a local approximation
- Not valid for large changes in the interest rate
- Relatedly, price‐interest rate relationship is “convex”" (point 4 in bond-pricing relationships slide)
Duration measures sensitivity to changes in yield to maturity
- Implicitly, this assumes shifts in the level of a flat yield curve
- Can be modified to accommodate different term structures
Application
Consider the case of Orange County, 1994
Bob Citron managed county funds as a large bet on interest rates
Borrowed large amounts in short term markets and reinvested in 5 year notes
Via leverage, portfolio had an effective duration of 7.4 years (2.74 duration of assets \(\times\) 2.7 leverage)
Orange County
Orange County
Orange County
Orange County
Orange County Application
Four consecutive rate hikes by the FOMC
The OC has a $1.6 billion loss on a $7.5 billion portfolio (before leverage), declares bankruptcy
Question: How does this 1.6BN loss compare to expectations?
A useful application of VaR and duration
Interest Rate Swaps
Fixed v. floating interest rate swap
One party pays a floating interest rate X notional value
Other party pays a fixed interest rate (the swap rate) X notional value
Exchanges made periodically over a fixed term
Fixed interest rate is set so that no cash exchanges hands at initiation
Swaps provide synthetic immunization
Interest Rate Swaps: Example
Note, counterparty which pays fixed and receives floating is short (bonds)
- Combining along bond portfolio with a short swap position is a means toward immunization
- e.g. add swap proceeds to a 7% coupon and you have a floating rate note
Pricing Swaps
Consider the following three period fixed vs. floating rate swap
Lending or borrowing at forward rates can be achieved synthetically, or in futures market
Consider cash flows from a hedged counterparty who received fixed and pays floating
Pricing Swaps
C is set to make the net present value of these streams equal to zero
This is known as the swap rate and is a weighted average of the spot and forward rates
\[ 0 = \frac{C- r_{0,1}}{(1+r_{1})} + \frac{C- f_{1,2}}{(1+r_{2})^{2}} + \frac{C- r_{2,3}}{(1+r_{3})^{3}} \]